by Dr. Carol JVF Burns (website creator)
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This section discusses the graph of the tangent function (shown below).
For ease of reference, some material is repeated from the Trigonometric Functions.

graph of $y = \tan x$
(periodic with period $\,\pi\,$)

The Tangent Function: Definition and Comments

Where does the graph of the tangent function come from?

Recall that every real number is uniquely defined by its sign (plus or minus) and its size (distance from zero).

Sign of the Tangent Function

The sign of $\,\tan t\,$ is easily determined from its definition as $\,\displaystyle\frac{\sin t}{\cos t}\,$:
  • Quadrant I: cosine positive, sine positive, tangent positive
  • Quadrant II: cosine negative, sine positive, tangent negative
  • Quadrant III: cosine negative, sine negative, tangent positive
  • Quadrant IV: cosine positive, sine negative, tangent negative

Sign of the Tangent Function

Size of the Tangent Function

The size of the tangent is best understood from its graphical significance
in the unit circle approach, as follows:
  • View the real number $\,t\,$ as the radian measure of an angle.
    Locate the angle $\,t\,$ in the unit circle.
    The terminal point for $\,t\,$ is shown in black at right.
  • Create an auxiliary (yellow) triangle (it overlaps the green triangle).
    The base of the yellow triangle has length $\,1\,.$
    The right-hand side is perpendicular to the $x$-axis and tangent to the circle.
  • The green and (overlapping) yellow triangles are similar.
    Why? They share the same central angle and they both have a right angle.
  • This chart gives lengths of sides in the green and yellow triangles:

     bottom of triangleright side of triangle
    green triangle:$\cos t$$\sin t$
    yellow triangle:$1$$y$
    (initially unknown)
  • By similarity: $$ \cssId{s52}{\frac{\sin t}{\cos t} = \frac{y}{1}}\,,\quad \cssId{s53}{\text{ so }}\quad \cssId{s54}{\color{blue}{y = \frac{\sin t}{\cos t} := \tan t}} $$
  • Thus, the size of the tangent gives the length of a (particular) segment that is tangent to the unit circle. (The word ‘tangent’ derives from the Latin tangens, meaning ‘touching’.)
  • Now, think about what happens to the length of this tangent segment as $\,t\,$ varies:
    As $\,t\,$ goes from to the size of $\,\tan t\,$ goes from to
    $-\frac{\pi}{2}$ $0$ $\infty$ $0$
    $0$ $\frac{\pi}{2}$ $0$ $\infty$
    $\frac{\pi}{2}$ $\pi$ $\infty$ $0$
    $\pi$ $\frac{3\pi}{2}$ $0$ $\infty$
    And so on!
  • Put the size and sign together to get the graph of the tangent function.

Size of the Tangent Function

Variations in Length
of the Tangent Segment

Size of the Tangent Function

Important Characteristics of the Graph of the Tangent Function

Tangent as Slope

There is an additional insight that comes from studying the diagram at right:
  • The length of the green segment is $\,1\,,$ since it is the radius of the unit circle.
  • The slope of the red line is therefore: $$ \cssId{s100}{\frac{\text{rise}}{\text{run}} = \frac{\tan t}{1} = \tan t} $$ The tangent gives the slope of the line!
  • This works in all the quadrants:

    in the second quadrant,
    $\text{slope} = \tan t\,$
    is negative

    in the third quadrant,
    $\text{slope} = \tan t\,$
    is positive

    in the fourth quadrant,
    $\text{slope} = \tan t\,$
    is negative
  • Summarizing: Start at the positive $x$-axis and lay off any angle $\,t\,$ (positive, negative, or zero). As long as the angle doesn't yield a vertical line (which has no slope), then $\,\tan t\,$ gives the slope of the resulting line. This is a useful fact in many applications.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
graph of the secant function
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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(MAX is 21; there are 21 different problem types.)
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